Geoscience Reference
In-Depth Information
Let us now assess the spatial scales at which the molecular destruction of enstro-
phy occurs. We again denote by
u
and
the scales of the energy-containing velocity
fluctuations. Let us scale the fluctuating vorticity as
ω
, the mean vorticity gradient
as
ω/
, and the fluctuating vorticity gradient as
ω/λ
2
, with the magnitude of the
length scale
λ
2
to be determined. Our scaling estimates for the leading terms in
Eq. (7.56)
are
ω
2
u
ν
ω
2
λ
2
,j
u
j
ω
∼
,
νω
,j
ω
,j
=
χ
ω
∼
,
so that equating the two yields
ν
u
1
/
2
ω
2
u
ν
ω
2
λ
2
∼
R
−
1
/
2
∼
λ
2
;
∼
.
(7.58)
t
When the turbulence Reynolds number
R
t
is large it follows from
(7.58)
that there
is a large scale range between
and
λ
2
, as in three-dimensional turbulence.
As with the Taylor microscale and viscous dissipation in three-dimensional tur-
bulence, we must be cautious in interpreting
λ
2
as the spatial scale of the eddies in
which the molecular destruction of enstrophy occurs. Rather,
ω/λ
2
is an estimate
of the vorticity gradients. Since the rms vorticity in the dissipative eddies is less
than
ω
, their length scale is less than
λ
2
.
7.5.3 Inertial-range cascades and scaling
Equation (7.58)
implies that large-
R
t
two-dimensional turbulence has a large range
of eddy scales
r
such that
λ
2
. These eddies contain little enstrophy and
do little molecular destruction of it. Interpreted for enstrophy in this inertial range
in two-dimensional turbulence, the scalar variance budget,
Eq. (7.4)
, indicates that
the enstrophy cascade rate is independent of wavenumber and numerically equal
to
χ
ω
, the mean rate of molecular destruction of enstrophy. If
(κ)
is the vorticity
spectrum,
r
∞
(κ)dκ
=
ω
2
,
(7.59)
0
then the
Kolmogorov
(
1941
) ideas imply that in this inertial subrange
χ
2
/
ω
κ
−
1
,
=
∼
(κ,χ
ω
)
(7.60)
as pointed out by
Kraichnan
(
1967
)and
Batchelor
(
1969
). Since the vorticity spec-
trum is
κ
2
times the energy spectrum
E(κ)
, it follows that the inertial-range energy
spectrum is
E(κ)
∼
χ
2
/
ω
κ
−
3
.
(7.61)
